Standard +0.3 This is a straightforward Further Maths integration question requiring completing the square to transform the integrand into the standard form 1/√(a²-x²), then applying the arcsin formula. While it involves Further Maths content (inverse trig integrals), the technique is routine and mechanical with no problem-solving required beyond recognizing the pattern. The exact value calculation is also standard.
# Question 3:
$\displaystyle\int_{\frac{1}{2}}^{1} \dfrac{1}{\sqrt{2x-x^2}}\,dx = \int_{\frac{1}{2}}^{1} \dfrac{1}{\sqrt{1+2x-x^2-1}}\,dx = \int_{\frac{1}{2}}^{1} \dfrac{1}{\sqrt{1-(1-x)^2}}\,dx$ | M1, A1 | Completing the square on given function
$= \left[-\sin^{-1}(1-x)\right]_{\frac{1}{2}}^{1}$ | M1, A1 | By substitution or using standard form where completed square is of form $1-(1\pm x)^2$; correct result of integration; ignore limits
$= -\left(0 - \dfrac{\pi}{6}\right) = \dfrac{\pi}{6}$ | A1 |
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